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Binary Message Passing Decoding of Product Codes Based on
Generalized Minimum Distance Decoding
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Citation for the original published paper (version of record):
Sheikh, A., Graell i Amat, A., Liva, G. (2019)
Binary Message Passing Decoding of Product Codes Based on Generalized Minimum Distance Decoding
53rd Annual Conference on Information Sciences and Systems (CISS). Invited paper
http://dx.doi.org/10.1109/CISS.2019.8692862
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(article starts on next page)Binary Message Passing Decoding of Product Codes
Based on Generalized Minimum Distance Decoding
Alireza Sheikh§ , Alexandre Graell i Amat§ , and Gianluigi Liva†
§
Department of Electrical Engineering, Chalmers University of Technology, Sweden
†
Institute of Communications and Navigation of the German Aerospace Center (DLR), Germany
(Invited Paper)
Abstract—We propose a binary message passing decoding algo- throughputs and energy consumption close to that of iBDD,
rithm for product codes based on generalized minimum distance another line of research recently explored is to improve the
decoding (GMDD) of the component codes, where the last stage performance of the conventional iBDD. In [10], an algorithm
of the GMDD makes a decision based on the Hamming distance
metric. The proposed algorithm closes half of the gap between that exploits conflicts between component codes in order
conventional iterative bounded distance decoding (iBDD) and to assess their reliabilities even when no channel reliability
turbo product decoding based on the Chase–Pyndiah algorithm, information is available, was proposed. The algorithm, dubbed
at the expense of some increase in complexity. Furthermore, the anchor decoding (AD), improves the performance of iBDD
proposed algorithm entails only a limited increase in data flow at the expense of some increase in decoding complexity. In
compared to iBDD.
[11], a decoding algorithm based on marking the least reliable
I. I NTRODUCTION bits was proposed for staircase codes. In [12], we proposed a
decoding algorithm based on BDD of the component codes,
Applications requiring very high throughputs, such as fiber- named iBDD with scaled reliability (iBDD-SR). The algorithm
optic communications and high-speed wireless communica- in [12] improves the performance of iBDD by exploiting chan-
tions, have recently triggered a significant amount of research nel reliabilities as proposed in [13] for LDPC codes, while still
on low-complexity decoders. While codes-on-graphs such as only exchanging binary (i.e., hard-decision) messages between
low-density parity-check (LDPC) codes and turbo codes have component decoders, similar to iBDD. iBDD-SR improves
been shown to provide close-to-capacity performance under upon iBDD and AD, and achieves the same throughput of
belief propagation (BP) decoding, scaling their BP decoders iBDD with a slight increase in energy consumption [14]. In
to yield throughtputs of the order of several Gbps or even [15], we proposed an algorithm based on generalized minimum
1 Tbps, as required for example for the the future optical distance decoding (GMDD) of the component codes. The
metro-networks, is a very challenging task. One of the main proposed algorithm closes over 50% of the performance gap
bottlenecks is the data flow required by the exchange of soft between iBDD and turbo product decoding (TPD) based on
information in the iterative BP decoding. This has spurred the Chase–Pyndiah algorithm [16], with lower complexity
a great deal of research in novel low-complexity decoding than TPD. However, the algorithm, which we referred to as
algorithms. iterative GMDD with scaled reliability (iGMDD-SR), requires
Several works have attempted to reduce the decoding com- the exchange of soft information between the component
plexity of BP decoding of LDPC codes, see, e.g., [1]–[4]. For decoders and hence entails a decoder data flow equivalent to
high-throughput applications, an alternative to LDPC codes that of TPD and significantly higher than that of iBDD.
with (BP) soft decision decoding (SDD) is to consider hard In this paper, we propose a novel binary message pass-
decision decoding (HDD). Product codes (PCs) [5], half- ing (BMP) decoding algorithm for product codes based on
product codes [6], staircase codes [7], braided codes [8], and GMDD of the component codes, which we refer to as BMP-
other product-like code structures [9] with HDD based on GMDD. The proposed algorithm follows the same principle
bounded distance decoding (BDD) of the component codes as the iGMDD-SR algorithm proposed in [15], but a crucial
(which we refer here to as iterative BDD (iBDD)) yield difference is that the Hamming distance metric is used at the
excellent performance with a significantly reduced data flow, final stage of the GMDD of the component codes. In contrast
hence achieving very high throughputs. However, this comes to iGMDD-SR, the resulting algorithm does not require the
at the expense of a performance loss (typically larger than 1 exchange of soft information, but the exchange of the hard
dB) compared to SDD. decisions on the code bits (as conventional iBDD) and an
To close the performance gap between iBDD of product-like ordered list of the dmin − 1 least reliable code bits for
codes and SDD of LDPC codes or product-like codes, yet with each component code, where dmin is the minimum Hamming
distance of the component code. This list can be represented
This work was financially supported by the Knut and Alice Wallenberg
Foundation, the Swedish Research Council under grant 2016-04253, and the by a small number of bits. The proposed algorithm yields
Ericsson Research Foundation. performance very close to that of iGMDD-SR, closing 50% ofc03 A. Generalized Minimum Distance Decoding
Consider the decoding of a BCH component code of length
n and the vector of channel LLRs l = (L1 , . . . , Ln ) corre-
c3 sponding to the received vector r = (r1 , . . . , rn ). GMDD is
c4 c4,3 c4
based on multiple algebraic error-erasure decoding attempts
[17]. In particular, the decoder ranks the coded bits in terms of
their reliabilities |L1 |, . . . , |Ln |. Then, the m least reliable bits
in r are erased, where m ∈ Modd , {dmin − 1, dmin − 3, ..., 2}
if dmin is odd and m ∈ Meven ,∈ {dmin − 1, dmin − 3, ..., 3}
Fig. 1. Code array (left) and simplified Tanner graph (right) of a PC with if dmin is even. For later use, we denote by L the ordered
identical component code of length n = 6 for row and column codes. In the
simplified Tanner graph, the CNs are represented by squares (the CNs on the list of dmin − 1 least reliable code bits. It can be readily
left represent the column codes and the CNs on the right represent the row checked that |Modd | = |Meven | = t, where t = dmin2−1
codes) and degree-2 VNs are represented as simple edges. The third column is the error correcting capability of the code. Together with
code and the fourth row code are highlighted.
the received vector r, this gives a list of t + 1 trial vectors
r̃i , i = 1, . . . , t + 1, out of which t vectors contain both
the performance gap between iBDD and TPD, while entailing erasures and (possibly) errors. Finally, algebraic error-erasure
only a small increase in data flow compared to iBDD (between decoding [18, Sec. 6.6] is applied to each trial vector r̃i ,
8.5% and 34.3%, depending on the code parameters). resulting in a set of candidate codewords, of size at most
Notation: We use boldface letters to denote vectors and t + 1, denoted by S . If decoding fails for all t + 1 vectors
matrices, e.g., x and X = [xi,j ]. The i-th row and j-th column in the list, a failure is declared. Otherwise, the decoder picks
of matrix X is denoted by Xi,: and X:,j , respectively. |a| among all candidate codewords in S the one that minimizes
denotes the absolute value of a, bac the largest integer smaller the generalized distance dGD (r, c), [17]
than or equal to a, and dae the smallest integer larger than or
equal to a. A Gaussian distribution with mean µ and variance ĉ = arg min dGD (r, c)
c∈S
σ 2 is denoted by N (µ, σ 2 ). X X
= arg min (1 − αi ) + (1 + αi ), (1)
c∈S i:ri =ci
II. P RELIMINARIES i:ri 6=ci
∆
Let C be an (n, k, dmin ) binary linear code, where n, where αi = |Li |/ max |Lj |. Note that if all input LLRs
1≤j≤n
k, and dmin are the code length, dimension, and minimum L1 , . . . , Ln have the same magnitude, we have αi = 1 for
distance, respectively. We consider two-dimensional PCs with all i = 1, . . . , n and (1) reverts to 2dH (r, ĉ), where dH (r, ĉ)
identical binary Bose–Chaudhuri–Hocquenghem (BCH) com- is the Hamming distance between r and ĉ.
ponent code C for the row and column codes. Such a PC, of By introducing erasures and performing multiple error-
parameters (n2 , k 2 , d2min ) and rate R = k 2 /n2 , is defined as erasure component decoding attempts, GMDD can decode
the set of all n × n arrays such that each row and each column beyond half the minimum distance of the code.
of the array is a codeword of C. Thus, a codeword of the PC
can be represented by an n × n binary matrix C = [ci,j ]. III. B INARY M ESSAGE PASSING D ECODING BASED ON
Alternatively, a PC can be defined via a Tanner graph with G ENERALIZED M INIMUM D ISTANCE D ECODING
2n constraint nodes (CNs), where n CNs correspond to the In this section, we propose a BMP decoding algorithm for
row codes and n CNs correspond to the column codes. The PCs based on GMDD of the component codes. We refer to it
graph has n2 variable nodes (VNs) corresponding to the n2 as BMP-GMDD. The algorithm follows the same principle as
code bits. The code array and (simplified) Tanner graph of a the iGMDD-SR algorithm that we proposed in [15]. However,
two-dimensional PC with n = 6 is shown in Fig. 1. compared to iGMDD-SR, the proposed BMP-GMDD does
We assume transmission over the binary-input additive not require the exchange of the reliabilities on the code bits
white Gaussian noise (AWGN) channel. The channel obser- between the row and column decoders. To achieve that, rather
vation corresponding to code bit ci,j is given by than considering the generalized distance in (1) to perform the
decision at the last stage of GMDD of the row and column
yi,j = xi,j + zi,j , decoders as in [15], we perform the decision based on the
Hamming distance, i.e., among all candidate codewords in S
where xi,j = (−1)ci,j , zi,j ∼ N (0, (2REb /N0 )−1 ), with
(see Section II-A), the decoder selects the one that minimizes
Eb /N0 being the signal to noise ratio. We denote by L = [Li,j ]
dH (r, c), i.e., the decision in (1) is substituted by
the matrix of channel log-likelihood ratios (LLRs) and by
R = [ri,j ] the matrix of hard decisions at the channel output, ĉ = arg min dH (r, c). (2)
where ri,j is obtained by mapping the sign of Li,j according c∈S
to 1 7→ 0 and −1 7→ 1. We denote this mapping by B(·), i.e., Making the decision based on the Hamming distance instead
ri,j = B(Li,j ). With some abuse of notation, we also write of the generalized distance entails a small performance loss, as
R = B(L). the decision does not take into consideration the normalizedreliabilities αi . However, this allows to significantly reduce code bit ci,j
i-th row r,(`)
wi Li,j j-th column
the decoder data flow, as explained later. c,(`−1) r,(`) r,(`) r,(`)
Ψi,: µ̄i,j ∈ {±1, 0} µi,j ψi,j
The proposed BMP-GMDD algorithm works as follows. GMDD × + B(·) GMDD
r,(`−1) c,(`)
Li Lj
Without loss of generality, assume that the decoding starts
with the row codes and let us consider the decoding of the
c,(`−1) Fig. 2. Block diagram showing the information flow from the i-th row decoder
i-th row code at iteration `. Let Ψc,(`−1) = [ψi,j ] be the to the j-th column decoder in BMP-GMDD. The message at the input of the
matrix of hard decisions on code bits ci,j after the decoding c,(`−1)
i-th row decoder is the vector of hard decisions on the code bits Ψi,: and
r,(`−1)
of the n column codes at iteration ` − 1. Also, let Li be the ordered list of the dmin − 1 least reliable bits Lj
r,(`−1)
from the decoding
the ordered list of dmin − 1 least reliable bits of codeword Ci,: of the column codes at the previous iteration.
from the decoding of the column codes at iteration ` − 1. Note
r,(`−1)
that in the first iteration the list Li is built according to
the ordering of the channel reliabilities Li,: = (Li,1 , . . . , Li,n ). column decoder, which entails a significantly higher decoder
Row decoding of the i-th row code is then performed based on data flow compared to BMP-GMDD.
c,(`−1) r,(`−1)
Ψi,: and Li . First, GMDD of the i-th row code based
c,(`−1)
on the Hamming distance is performed based on Ψi,: and IV. D ECODING C OMPLEXITY D ISCUSSION
Lri , as explained in Section II-A (see (2)). Note that GMDD
does not provide reliability information about the decoded bits, A thorough complexity analysis of BMP-GMDD should
i.e., it is a soft-input hard-output decoding algorithm. In order include, besides pure algorithmic aspects, implementation
to provide the column decoders with the list of m least reliable implications in terms of memory requirements, wiring, and
bits for each codeword C:,j after the decoding of the row transistor switching activity [14], and is beyond the scope of
codes at iteration `, we do the following. The output bits of this paper. We however provide a high-level discussion of the
GMDD are mapped according to 0 7→ +1 and 1 7→ −1 if complexity and data flow of BMP-GMDD compared to that of
GMDD is successful and mapped to 0 if GMDD fails. Let conventional iBDD, AD [10], iBDD-SR [12], and iGMDD-SR
r,(`)
µ̄i,j ∈ {±1, 0} be the result of this mapping for the decoded [15].
bit corresponding to code bit ci,j . The reliability information Conventional iBDD, iBDD-SR, and AD are based on BDD
is then formed according to of the component codes and are characterized by a similar
r,(`) r,(`) r,(`) complexity and data flow. In particular, it was shown in [14]
µi,j = wi · µ̄i,j + Li,j , (3)
that for the same data throughput (up to 1 Tbps), iBDD-SR
where wi
r,(`)
> 0 is a scaling factor than needs to be optimized. provides 0.2–0.25 dB gain with respect to iBDD with only
Then, the hard decision on ci,j made by the i-th row decoder slightly higher energy consumption.
is formed as Both GMDD-SR and the proposed BMP-GMDD are based
r,(`) r,(`)
ψi,j = B(µi,j ). (4) on GMDD of the component codes. In this case, t error-
erasure decoding attempts and one BDD attempt are required.
r,(`)
The hard decision ψi,j is the binary message on code bit Each error-erasure decoding attempt has a cost close to a run
ci,j passed from the i-th row code to the j-th column code, of BDD. Each decoding attempt may result in a candidate
i.e., from the i-th row CN to the j-th column CN (see Fig. 1). codeword that is used to form a list of size up to t + 1, as
In particular, after applying this procedure to all row codes, explained in Section II-A. The minimization of the generalized
r,(`)
the matrix Ψr,(`) = [ψi,j ] is formed and used as the input for distance in (1) for GMDD-SR and the Hamming distance in
the n column decoders. Furthermore, after decoding of all row (2) for BMP-GMDD has a negligible cost with respect to the
codes, for each column codeword C:,j , the corresponding code t+1 decoding attempts. On the other hand, both BMP-GMDD
r,(`) r,(`) and iGMDD-SR require finding the dmin − 1 least reliable bits
bits are ranked according to the reliabilities (µ1,j , . . . , µn,j ).
c,(`) and sorting them according to their reliabilities, which adds
Then the m least reliable bits are stored in the list Lj , which some further complexity.
is passed to the j-th column decoder.
Note that GMDD-SR requires the component decoders to
The decoding of the n column codes at iteration ` is then be provided with soft information by the previous decoding
performed based on the hard decisions Ψr,(`) and the lists of iteration. Therefore, its data flow is significantly higher than
c,(`) c,(`)
least reliable bits L1 , . . . , Ln as explained for the i-th that of iBDD, iBDD-SR, and AD, and is the same of soft deci-
row decoder above. After decoding of the n column codes sion TPD. For an a-bit representation of the soft information,
c,(`)
at decoding iteration `, the matrix Ψc,(`) = [ψi,j ] of hard the data flow is roughly a times that of BDD, iBDD-SR, and
r,(`) r,(`)
decision bits and the lists L1 , . . . , Ln are passed to the AD. In contrast, BMP-GMDD requires only the exchange of
n row decoders for the next decoding iteration. The iterative the hard decisions and the ordered list of dmin −1 least reliable
process continues until a maximum number of iterations is bits for each row and column codeword. For a component code
reached. The BMP-GMDD of PCs is schematized in Fig. 2. of length n, the index of each code bit can be represented
Remark: With reference to Fig. 2, the iGMDD-SR algorithm with dlog2 (n)e bits. Furthermore, for each of the dmin − 1
r,(`)
proposed in [15] passes the soft information µi,j to the j-th least reliable bits we need to provide their ordering in termsTable I
C OMPARISON OF DIFFERENT PRODUCT DECODING ALGORITHMS . C ODING GAINS AND CAPACITY GAPS ARE MEASURED AT A BER OF 10−6
channel exchanged gain over gap from
acronym decoding algorithm
reliabilities messages iBDD (dB) capacity (dB)
iBDD iterative bounded distance decoding no hard - 1.03 (HD)
iBDD (ideal) iterative bounded distance decoding without miscorrections no hard 0.28 0.75 (HD)
iBDD-SR iterative bounded distance decoding with scaled reliability [12] yes hard 0.27 2.3 (SD)
AD anchor decoding [10] no hard 0.18 0.85 (HD)
BMP-GMDD binary message passing decoding based on GMD decoding yes hard 0.51 1.79 (SD)
iGMDD-SR iterative generalized minimum distance decoding with scaled reliability [15] yes soft 0.58 1.72 (SD)
TPD turbo product decoding (Chase–Pyndiah) [16] yes soft 1.08 1.22 (SD)
10−1 [15], and TPD based on the Chase-Pyndiah decoding [16]. For
all algorithms, a maximum of `max = 10 decoding iterations
10−2 HD ca is performed. As a reference, the Shannon limit for SDD and
pacity
HDD is also plotted in the figure.
10−3 Both BMP-GMDD and iGMDD-SR require a proper choice
(`)
SD capacity
of the scaling factors wi . For simplicity, we consider the
BER
10−4 same scaling factor for all row and column codes, i.e.,
iBDD r,(`) c,(`)
wi = wj = w(`) for all i, j = 1, . . . , n, and jointly
AD
10−5 ideal iBDD optimize the vector w = (w(1) , . . . , w(`max ) ) by using Monte–
iBDD-SR Carlo estimates of the BER for a fixed Eb /N0 as the op-
iGMDD-SR timization criterion. Intuitively, one would expect that the
10−6
BMP-GMDD decisions of the component decoders become more reliable
TPD
with increasing number of iterations, whereas the channel
10−7
2.5 3 3.5 4 4.5 5 5.5 observations become less informative. Therefore, in order
Eb /N0 (dB) to reduce the optimization search space, we only consider
vectors w with monotonically increasing entries. iBDD-SR
Fig. 3. BER performance of different decoding algorithms for a PC with
(256, 239, 6) eBCH component codes and transmission over the AWGN also requires scaling factors (see [12], [19]). In this case, the
channel. The PC rate is 0.8716 and the maximum number of decoding scaling factors can be derived using density evolution [13],
iterations is 10. [19].
The two reference curves are conventional iBDD (red curve
r,(`) with empty triangle markers) and TPD (purple curve with
of reliabilities. Thus, each ordered lists Li , i = 1, . . . , n,
c,(`)
and Lj , = j, . . . , n, can be represented with pentagon markers), with the latter performing 1.1 dB better
at a BER of 10−7 . AD (dark blue curve with filled circle
(dlog2 (n)e + dlog2 (dmin − 1)e) (dmin − 1) markers) and iBDD-SR (pink curve with filled triangle mark-
ers) outperform conventional iBDD by 0.18 dB and 0.27 dB,
bits each. This is the additional data flow (per row and column respectively, at the same BER. As a reference, we also show
code decoding) compared to conventional iBDD. For instance, the performance of ideal iBDD (brown curve with empty circle
for a component code of code length n = 256 bits, the data markers), where a genie prevents miscorrections. Interestingly,
flow of BMP-GDD is 15.625% and 34.375% higher than that at a BER of 10−7 iBDD-SR yields the same performance as
of iBDD for dmin = 5 (t = 2) and dmin = 9 (t = 4), ideal iBDD.1 iGMDD-SR (green curve with diamond markers)
respectively. For a component code of length n = 512 bits, outperforms iBDD, iBDD-SR, and AD and closes ≈ 54% of
the increase in data flow is reduced to 8.593% and 18.75%, the gap between iBDD and TPD, at the expense of an increased
respectively. Thus, the increase in data flow of BMP-GMDD complexity and data flow.
compared to iBDD is very limited and is much lower than the
The performance of the proposed BMP-GMDD is given by
data flow of iGMDD-SR and conventional TPD.
the blue curve with square markers. The proposed decoding
V. N UMERICAL R ESULTS algorithm yields performance very close to that of iGMDD-SR
(a performance degradation compared to iGMDD-SR of only
In Fig. 3, we give the bit error rate (BER) performance
0.074 dB is observed at a BER of 10−7 ), while achieving
of BMP-GMDD for a PC with double-error-correcting ex-
a significantly lower data flow. BMP-GMDD closes around
tended BCH (eBCH) codes with parameters (256, 239, 6) as
component codes and transmission over the AWGN channel.
1 We remark that the performance of iBDD-SR in Fig. 3 is improved
The resulting PC has rate R = 2392 /2562 ≈ 0.8716. For
compared to [15], since in this paper we use the optimized scaling factors
comparison purposes, we we also plot the performance of based on the density evolution derived in [19], rather than based on Monte-
conventional iBDD, AD [10], iBDD-SR [12], iGMDD-SR Carlo simulations as in [15].50% of the performance gap between iBDD and TPD, while staircase codes. Overall, the proposed BMP-GMDD algorithm
requiring only a 21.48% higher data throughput than iBDD. provides a very good performance-complexity tradeoff and is
The coding gain improvements of all considered decoding appealing for very high-throughput applications such as fiber-
algorithms over iBDD are summarized in Table I (fifth col- optic communications.
umn). In the table we also indicate whether the algorithms ex-
ACKNOWLEDGMENT
ploit the channel reliabilities or not, the nature of the messages
exchanged in the iterative decoding (hard or soft), as well as The authors would like to thank Dr. Christian Häger for
the gap to capacity for all schemes (sixth column). Note that providing the simulation results of anchor decoding in Fig. 3.
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